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Theorem funoprab 6745
Description: "At most one" is a sufficient condition for an operation class abstraction to be a function. (Contributed by NM, 17-Mar-1995.)
Hypothesis
Ref Expression
funoprab.1 ∃*𝑧𝜑
Assertion
Ref Expression
funoprab Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
Distinct variable group:   𝑥,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem funoprab
StepHypRef Expression
1 funoprab.1 . . 3 ∃*𝑧𝜑
21gen2 1721 . 2 𝑥𝑦∃*𝑧𝜑
3 funoprabg 6744 . 2 (∀𝑥𝑦∃*𝑧𝜑 → Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑})
42, 3ax-mp 5 1 Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
Colors of variables: wff setvar class
Syntax hints:  wal 1479  ∃*wmo 2469  Fun wfun 5870  {coprab 6636
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-br 4645  df-opab 4704  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-fun 5878  df-oprab 6639
This theorem is referenced by:  mpt2fun  6747  ovidig  6763  ovigg  6766  oprabex  7141  axaddf  9951  axmulf  9952  funtransport  32113  funray  32222  funline  32224
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